where \(A(x)\equiv \eta ^{\mu \nu }A_{\mu \nu }(x)\)  and \(\eta _{\mu \nu }=\mbox{diag}(-1,1,1,1)\)  is the 4D Minkowski metric.
The action (REF ) depends on a dimensionless parameter, which we called “\(a\) ”. It is quite peculiar that a free quadratic theory shows a “coupling” constant which cannot be reabsorbed by a field redefinition. To our knowledge, the action (REF ) is the only non interacting theory displaying this feature, with the exception of the abelian 3D Maxwell-Chern-Simons theory, where the coupling plays the role of a topological mass [1]}. The choice of the \(a\) -parametrization of the action (REF ) makes apparent the contact with LG [2]}, which is reached for \(a=0\)  :
\(S_{inv}(a=0)=S_{LG}\ ,\) 
